4. introduction and chapter objectives - digilentincreal analog - circuits 1 chapter 4: systems and...

Real Analog - Circuits 1Chapter 4: Systems and Network Theorems

© 2012 Digilent, Inc. 1

4. Introduction and Chapter ObjectivesIn previous chapters, a number of approaches have been presented for analyzing electrical circuits. In theseanalysis approaches, we have been provided with a circuit consisting of a number of elements (resistors, powersupplies, etc.) and determined some circuit variable of interest (a voltage or current, for example). In the processof determining this variable, we have written equations which allow us to determine any and all variables in thesystem. For a complex circuit, with many elements, this approach can result in a very large number of equationsand a correspondingly large amount of effort expended in the solution of these equations. Unfortunately, much ofthe physical insight about the overall operation of the circuit may be lost in the detailed description of all of theindividual circuit elements. This limitation becomes particularly serious when we attempt to design a circuit toperform some task.

In this chapter, we introduce the concept of a systems level approach to circuit analysis. In this type of approach,we represent the circuit as a system with some inputs and outputs. We then characterize the system by themathematical relationship between the system inputs and the system outputs. This relationship is called theinput-output relation for the system. This representation of a system leads to several network theorems whose usecan simplify the analysis of these systems. The network theorems essentially allow us to model a portion of acomplicated circuit as a much simpler (but equivalent) circuit. This simplified model can then be used to facilitatethe design or analysis of the remainder of the circuit.

The above approach for representing circuits is particularly useful in circuit design; successful design approachesfor large circuits typically use a top-down strategy. In this design approach, the overall system is broken down intoa number of interconnected subsystems, each of which performs some specific task. The input-outputrelationships for these individual subsystems can be determined based on the task to be performed. Thesubsystems can then be designed to implement the desired input-output relation. An audio compact disc player,for example, will include subsystems to perform filtering, digital-to-analog conversion, and amplificationprocesses. It is significantly easier to design the subsystems based on their individual requirements than toattempt to design the entire system all at once. We will thus begin to think of the circuits we analyze as systemswhich perform some overall task, rather than as collections of individual circuit elements.

Real Analog – Circuits 1Chapter 4: Systems and Network Theorems


After completing this chapter, you should be able to:

Define signals and systems Represent systems in block diagram form Identify system inputs and outputs Write input-output equations for systems State the defining properties of linear systems Determine whether a system is linear State conditions under which superposition can be applied to circuit analysis Analyze electrical circuits using the principle of superposition Define the i-v characteristic for a circuit Represent a resistive circuit in terms of its i-v characteristic Represent a resistive circuit as a two-terminal network Determine Thévenin and Norton equivalent circuits for circuits containing power sources and resistors Relate Thévenin and Norton equivalent circuits to i-v characteristics of two-terminal networks Determine a load resistance which will maximize the power transfer from a circuit

Real Analog – Circuits 1Chapter 4.1: Signals and Systems


4.1: Signals and SystemsIn this section, we introduce basic concepts relative to systems-level descriptions of general physical systems.Later sections will address application of these concepts specifically to electrical circuits.

A system is commonly represented as shown in the block diagram of Figure 4.1. The system has some input, u(t),and some output, y(t). In general, both the input and output can be functions of time; the case of constant valuesis a special case of a time-varying function. The output will be represented as some arbitrary function of the input:

)()( tufty (4.1)

Equation (4.1) is said to be the input-output equation governing the system. The above relationship has only oneinput and one output – the system is said to be a single-input-single-output (SISO) system. Systems can havemultiple inputs and multiple outputs, in these cases there will be an input-output equation for each system outputand each of these equations may be a function of several inputs. We will concern ourselves only with SISOsystems for now.

Figure 4.1. Block diagram representation of a system.

One important aspect of the systems-level approach represented by equation (4.1) and Figure (4.1) is that we arerepresenting our system as a “black box”. We really have no idea what the system itself is, beyond a mathematicaldependence of the output variable on the input variable. The physical system itself could be mechanical, thermal,electrical, or fluidic. In fact, it is fairly common to represent a mechanical system as an “equivalent” electricalsystem (or vice-versa), if doing so increases the physical insight into the system’s operation.

The circuits we analyze can now be thought of as systems which perform some overall task, rather than ascollections of individual circuit elements. We will also think of the inputs and outputs of the system as signals,rather than specific circuit parameters such as voltages or currents. This approach is somewhat more abstractthan we are perhaps used to, so we will provide some additional discussion of what we mean by these terms.

Generally, most people think of a system as a group of interrelated “elements” which perform some task. Thisviewpoint, though intuitively correct, is not specific enough to be useful from an engineering standpoint. In thesechapters, we will define a system as a collection of elements which store and dissipate energy. The systemtransfers the energy in the system inputs to the system outputs; the process of energy transfer is represented by theinput-output equation for the system. Examples of the energy transfer can include mechanical systems (thekinetic energy resulting from using a force to accelerate a mass, or the potential energy resulting from using aforce to compressing a spring), thermal systems (applying heat to change a mass’s temperature), and electricalsystems (dissipating electrical power with the filament in a light bulb to produce light).

The task to be performed by the system of Figure 4.1 is thus the transformation of some input signal u(t) into anoutput signal y(t). Signals, for us, will be any waveform which can vary as a function of time. This is an extremelybroad definition – examples of signals include:



the force applied to a mass, the velocity of the mass as it accelerates in response to the applied force, the current applied to a circuit by a power supply, the voltage difference across a resistor which is subjected to some current flow, the electrical power supplied to a heating element, the temperature of a mass which is being heated by an electric coil

The transformation of the input signal to the output signal is performed by the input-output relation governingthe system. The input-output relation can be a combination of algebraic, differential, and integral equations.

To provide some concrete examples of the above concepts, several examples of systems-level representations ofcommon processes are provided below.

Example 4.1: Mass subjected to an external force

Consider the mass-damper system shown in the figure below. The applied force F(t) pushes the mass to the right.The mass’s velocity resulting from the applied force is v(t). The mass slides on a surface with sliding coefficient offriction b, which induces a force Fb = bv(t) which opposes the mass’s motion. The mass is initially at rest and theapplied force is zero for time before time t=0.

The governing equation for the system (obtained by drawing a free body diagram of the mass and applying

maF ) is

)()()( tFtbvdttdvm

The governing equation for the system is a first order differential equation. Knowledge of the externally appliedforce F(t) and the initial velocity of the mass allows us to determine the velocity of the mass at all subsequenttimes. Thus, we can model the system as having an input signal F(t) – which is known – and an output velocityv(t) which can be determined from the input signal and the properties of the system (the mass, m, and coefficientof friction, b). The system can then be represented by the block diagram below:

)()()( tFtbvdttdvm

(It is rather unusual to place the system governing equation directly in a block diagram; we do it here to illustrate apoint.)



Example 4.2: Electrical circuit

For the electrical circuit below, write the equations governing the input-output relationship for the circuit. Theapplied input to the circuit is the voltage source Vin and the output is the voltage V across the 2 resistor.

We previously wrote mesh equations for this circuit (for a specific value of Vin) in Chapter 3.2. We repeat thesemesh equations here, along with our definitions of the mesh currents:

0)3(24 11 inX VVii

05)3(3 22 iViV Xin

25iVX

The output voltage V is related to the mesh currents by:

)3(2 1 xViV

The above four equations provide an input-outputdescription of the circuit. If desired, they can becan be combined to eliminate all variables exceptVin and V and re-written in the form inVfV per equation (1). Note that all information about the original system, except the relationship between the inputand output signals, is lost once we do this.

The system-level block diagram for the circuit might then be drawn as:



Example 4.3: Temperature control system

Our final example is of a temperature control system. This example illustrates the representation of a complexsystem as a set of interacting subsystems.

A typical temperature control system for a building will have a thermostat which allows the occupants to set adesired temperature, a furnace (or air conditioner) which provides a means of adjusting the building’stemperature, some way of measuring the actual building temperature, and a controller which decides whether toturn the furnace or air conditioner on or off, based on the difference between the desired and actual temperatures.The block diagram below provides one possible approach toward interconnecting these subsystems into an overalltemperature control system. This block diagram can be used to identify individual subsystems, and providespecifications for the subsystems, which can allow the design to proceed efficiently. For example:

1. The temperature measurement system might be required to produce a voltage, which is a function of thetemperature in the building. The thermistor-based temperature measurement systems we have designed andconstructed in the lab are good examples of this type of system.

2. The controller might operate by comparing the desired temperature (generally represented by a voltage level)with the voltage indicating the actual temperature. For a heating system, if the actual temperature is lowerthan desired by some minimum amount, the controller will make a decision to switch the furnace on. Designdecisions might be made to determine what minimum temperature difference is required to turn the furnaceon, and whether to base the decision to turn on the furnace strictly upon a temperature difference or on a rateof change in temperature difference.

3. When the furnace turns on it will apply heat to the building, causing the building’s temperature to increase.Once the building temperature is high enough, the controller will then typically turn the furnace back off.The furnace must be designed to provide appropriate heat input to the building, based on the building sizeand the anticipated heat losses to the building’s surroundings. (For example, a larger building or a building ina colder climate will require a larger furnace.)

4. A model of the building’s heat losses will generally be necessary in order to size the furnace correctly andchoose an appropriate control scheme. Design choices for the building itself may include insulationrequirements necessary to satisfy desired heating costs.

Designs for the above subsystems can now proceed somewhat independently, with proper coordination betweenthe design activities.



Section Summary:

Systems are a set of components which work together to perform some task. Systems are typically consideredto have one or more inputs (which are provided to the system from the external environment) and one ormore outputs (which the system provides to the environment).

Generically, the inputs and outputs of systems are signals. Signals are simply time-varying functions. Theycan be voltages, currents, velocities, pressures, etc.

Systems are often characterized by their input-output equations. The input-output equation for a systemsimply provides a mathematical relationship between the input to the system and the output from the system.Once the input is defined as a particular number or function of time, that value or function can be substitutedinto the input-output equation to determine the system’s response to that input.

Exercises:

1. The input to the circuit below is the current, U. The output is the current through the 10Ω resistor, I.Determine an input-output equation for the circuit.

2. The input to the circuit is the voltage U. The output is the voltage V1. Determine an input-output relationfor the circuit.

41A

+

-U

4 +

-

V1

Real Analog – Circuits 1Chapter 4.2: Linear Systems


4.2: Linear SystemsWe have so far introduced a number of approaches for analyzing electrical circuits, including: Kirchoff’s currentlaw, Kirchoff’s voltage law, circuit reduction techniques, nodal analysis, and mesh analysis. When we have appliedthe above analysis methods, we have generally assumed that any circuit elements operate linearly. For example,we have used Ohm’s law to model the voltage-current relationships for resistors. Ohm’s law is applicable only forlinear resistors – that is, for resistors whose voltage-current relationship is a straight line described by the equationv = Ri. Non-linear resistors have been mentioned briefly; in lab assignment 1, for example, we forced a resistor todissipate an excessive amount of power, thereby causing the resistor to burn out and display nonlinear operatingcharacteristics. All circuit elements will display some degree of non-linearity, at least under extreme operatingconditions.

Unfortunately, the analysis of non-linear circuits is considerably more complicated than analysis of linear circuits.Additionally, in subsequent chapters we will introduce a number of analysis methods which are applicable only tolinear circuits. The analysis of linear circuits is thus very pervasive – for example, designing linear circuits is muchsimpler than the design of non-linear circuits. For this reason, many non-linear circuits are assumed to operatelinearly for design purposes; non-linear effects are accounted for subsequently during design validation andtesting phases.

The concept of treating an electrical circuit as a system was introduced in section 4.1. In systems-level analysis ofcircuits, we are primarily interested in the relationship between the system’s input and output signals. Circuitsgoverned by nonlinear equations are considered to be nonlinear systems; circuits whose governing input-outputrelationship is linear are linear systems. In this chapter, we formally introduce the concept of linear systems. Theanalysis of linear systems is extremely common, for the reasons mentioned above: structural systems, fluiddynamic systems, and thermal systems are often analyzed as linear systems, even though the underlying processesare often inherently nonlinear. Linear circuits are a special case of linear systems, in which the system consistsonly of interconnected electrical circuit elements whose voltage-current relationships are linear.

Linear systems are described by linear relations between dependent variables. For example, the voltage-currentcharacteristic of a linear resistor is provided by Ohm’s law:

iRv

where v is the voltage drop across the resistor, i is the current through the resistor, and R is the resistance of theresistor. Thus, the dependent variables – current and voltage – are linearly related. Likewise, the equations wehave used to describe dependent sources (provided in section 1.2):

Voltage controlled voltage source: 1vvs Voltage controlled current source: 1gvis Current controlled voltage source: 1rivs Current controlled current source: 1iis

all describe linear relationships between the controlled and controlling variables.



All of the above relationships are of the form

)()( tKxty (4.2)

where x(t) and y(t) are voltages or currents in the above examples. More generally, x(t) and y(t) can be consideredto be the input and output signals, respectively, of a linear system. Equation 4.2 is often represented in blockdiagram form as shown in Figure 4.2.

Figure 4.2. Linear system block diagram.

The output is sometimes called the response of the system to the input. The multiplicative factor K relating theinput and output is often called the system’s gain. Elements which are characterized by relationships of the formof equation 4.2 are sometimes called linear elements. The equation relating the system’s input and outputvariables is called the input-output relationship of the system.

Aside:

Many types of systems can be described by the relationship of equation (1). For example, Hooke’s law, whichrelates the force applied to a spring to the spring’s displacement, is

xkF

where k is the spring constant, F is the applied force, and x is the resulting displacement as shown below. In thisexample, F is the input to the system and x is the system output.

Notice that we have allowed the input and output of our system to vary as functions of time. Constant values arespecial cases of time-varying functions. We will assume that the system gain is not a time-varying quantity.

For our purposes, we will define linearity in somewhat more broad terms than equation (4.2). Specifically, we willdefine a system as linear if it satisfies the following requirements:



Linearity:

1. If the response of a system to some input x1(t) is y1(t) then the response of the system to some input x1(t)is y1(t), where is some constant. This property is called homogeneity.

2. If the response of the same system to an input x2(t) is y2(t), then the response of the system to an inputx1(t)+x2(t) is y1(t)+y2(t). This is called the additive property.

The above two properties defining a linear system can be combined into a single statement, as follows: if theresponse of a system to an input x1(t) is y1(t) and the system’s response to an input x2(t) is y2(t), then the responseof the system to an input x1(t)+x2(t) is y1(t)+y2(t). This property is illustrated by the block diagram of Figure4.3. The symbol in Figure 4.3 denotes signal summation; the signs on the inputs to the summation blockindicate the signs to be applied to the individual signals.

Figure 4.3. Block diagram representation of properties defining a linear system.

The above definition of linearity is more general than the expression of equation (4.2). For example, the processesof differentiation and integration are linear processes according to the above definition. Thus, systems with theinput-output relations such as:

xdtay anddtdxby

are linear systems. We will use circuit elements which perform integrations and differentiations later when wediscuss energy storage elements such as capacitors and inductors.

Dependent Variables and Linearity:

Linearity is based on the relationships between dependent variables, such as voltage and current. In order for asystem to be linear, relationships between dependent variables must be linear – plots of one dependent variableagainst another are straight lines. This causes confusion among some readers when we begin to talk about timevarying signals. Time is not a dependent variable, and plots of voltages or currents as a function of time for alinear system may not to be straight lines.

Although the above definitions of linear systems are fundamental, we will not often use them directly. Kirchoff’svoltage law and Kirchoff’s current law rely upon summing multiples of voltages or currents. As long as thevoltage-current relations for individual circuit elements are linear, application of KVL and KCL to the circuit will



result in linear equations for the system. Therefore, rather than direct application of the above definitions oflinear systems, we will simply claim that an electrical circuit containing only linear circuit elements will be linearand will have linear input-output relationships. All circuits we have analyzed so far have been linear.

Linearity:

If all elements in a circuit have linear voltage-current relationships, the overall circuit will be linear.

Important note about power:

A circuit’s power is not a linear property, even if the voltage-current relations for all circuit elements are linear.

Resistors which obey Ohm’s law dissipate power according to RiRvivP 22

. Thus, the power dissipation of

a linear resistor is not a linear combination of voltages or currents – the relationship between voltage or currentand power is quadratic. Thus, if power is considered directly in the analysis of a linear circuit, the resulting systemis nonlinear.

Section Summary:

Linear systems are characterized by linear relationships between dependent variables in the system. Forelectrical system, this typically means that the relationship between voltage and current for any circuitcomponent is linear – in electrical circuits, for example, this means that a plot of voltage vs. current for everyelement in the system is a straight line. Ohm’s law, for example, describes a linear voltage-currentrelationship.

Linear systems have a very important property: the additive principle applies to them. Superpositionessentially means that the response of a system to some combination of inputs x1+x2 will be the same as thesum of the response to the individual inputs x1 and x2.



Excercises:

1. The 20Ω resistor below obeys Ohm’s law, so that V=20I. We will consider the input to be the current throughthe resistor and the output to be the voltage drop across the resistor. Determine:

a. The output V if the input I=2A.b. The output V if the input I = 3A.c. The output V is the input I = 2A + 3A = 5A.

Do your answers above indicate that the additive property holds for this resistor? Why?

2. A linear electrical circuit has an input voltage V1 and provides an output voltage V2, as indicated in the blockdiagram below. If an input voltage V1 = 3V is applied to the circuit, the measured output voltage V2 = 2V.What is the output voltage if an input voltage V1 = 6V is applied to the circuit?

Real Analog – Circuits 1Chapter 4.3: Superposition


4.3: SuperpositionIn section 4.2, we stated that, by definition, the input-output relations for linear systems have an additive property.The additive property of linear systems states that:

If the response of a system to an input x1(t) is y1(t) and the response of the system to an input x2(t) is y2(t),then the response of the system to an input x1(t)+ x2(t) is y1(t)+2(t).

Thus, if a system has multiple inputs, we can analyze the system’s response to each input individually and thenobtain the overall response by summing the individual contributions. This property can be useful in the analysisof circuits which have multiple sources. If we consider the sources in a circuit to be the inputs, linear circuits withmultiple independent sources can be analyzed by determining the circuit’s response to each source individually,and then summing, or superimposing, the contributions from each source to obtain the overall response of thecircuit to all sources. In general, the approach is to analyze a complicated circuit with multiple sources bydetermining the responses of a number of simpler circuits – each of which contains only a single source.

We illustrate the overall approach graphically by the block diagram of Figure 4.4 (which is really just a reversedform of the block diagram of Figure 4.3).

Figure 4.4. Additive property of linear systems.

In Figure 4.4, we have a linear system with two input signals which are applied by sources in the circuit. We cananalyze this circuit by noting that each input signal corresponds to an independent source in the circuit. Thus, ifthe circuit’s overall response to a source x1(t) is y1(t) and the circuit’s response to a source x2(t) is y2(t), then thetotal circuit response will be the sum of the two individual responses, y1(t) + y2(t). Thus, if we wish to determinethe response of the circuit to both sources, x1(t) and x2(t), we can determine the individual responses of the circuit,y1(t) and y2(t) and then sum (or superimpose) the responses to obtain the circuit’s overall response to both inputs.This analysis method is called superposition.

In order to determine a circuit’s response to a single source, all other independent sources must be turned off (or,in more colorful terminology, killed, or made dead). To turn off a current source, we must make the input currentzero, which corresponds to an open circuit. To turn off a voltage source, we must make the input voltage zero,which corresponds to a short circuit.

Killing Sources:

To kill a voltage source, replace it with a short circuit To kill a current source, replace it with an open circuit.



To apply the superposition method, then, the circuit’s response to each source in the circuit is determined, with allother sources in the circuit dead. The individual responses are then algebraically summed to determine the totalresponse to all inputs. To illustrate the method, we consider the examples below.

Example 4.4:

Determine the voltage V in the circuit below, using superposition.

The circuit above can be consider to be the superposition of the two circuits shown below, each with a singlesource (the other source, in both cases, has been killed).

The voltage V1 above can be determined to be the result of a current division: VAV 22321

11

.

V2 can be determined to be the result of a voltage division: VVV 812

2121

. Thus, the voltage

VVVV 1021 .



Example 4.5:

Determine the voltage V in the circuit below, using superposition.

We begin by determining the response V1 to the 6V source by killing the 2A source, as shown in the figure below.

The voltage V1 is simply the result of a voltage division: VVV 2631

1 . The response V2 to the 2A source can

be determined by killing the 6V source, resulting in the circuit below:

Killing the 6V source places a short circuit in parallel with the 2A source, so no voltage is induced in any of theresistors by the 2A source. Thus, V2 = 0V.

The voltage V is the sum of the two individual voltages: V = V1 + V2 = 2V + 0V = 2V.



Notes on Superposition:

1. Superposition cannot be used directly to determine power. Previously, we noted that power is notgoverned by a linear relationship. Thus, you cannot determine the power dissipated by a resistor bydetermining the power dissipation due to each source and then summing the results. You can, however,use superposition to determine the total voltage or current for the resistor and then calculate the powerfrom the voltage and/or current.

2. When using superposition to analyze circuits with dependent sources, do not kill the dependent sources.You must include the effects of the dependent sources in response to each independent source.

3. Superposition is a powerful circuit analysis tool, but its application can result in additional work. Beforeapplying superposition, examine the circuit carefully to ensure that an alternate analysis approach is notmore efficient. Circuits with dependent sources, in particular, tend to be difficult to analyze usingsuperposition.

Section Summary:

Superposition is a defining property of linear systems. It essentially means that, for linear systems, we candecompose any input to the system into a number of components, determine the system output resultingfrom each component of the input, and obtain the overall output by summing up these individualcomponents of the output.

Superposition can be used directly to analyze circuits which contain multiple independent sources. Theresponses of the circuit to each source (killing all other sources) are determined individually. The overallresponse of the circuit – due to all sources – is then obtained by summing (superimposing) these individualcontributions.

The principle of superposition is a fundamental property of linear systems and has very broad-rangingconsequences. We will be invoking it throughout the remainder of this textbook, often without overtly statingthat superposition is being used. The fact that superposition applies to linear circuits is the basic reason whyengineers make every possible attempt to use linear models when analyzing and designing systems.

Exercises:

1. Use superposition to determine the voltage V1 in the circuit below.

Real Analog – Circuits 1Chapter 4.4: Two-terminal Networks


4.4: Two-terminal NetworksAs noted in section 4.1, it is often desirable, especially during the design process, to isolate different portions of acomplex system and treat them as individual subsystems. These isolated subsystems can then be designed oranalyzed somewhat independently of one another and subsequently integrated into the overall system in a top-down design approach. In systems composed of electrical circuits, the subsystems can often be represented astwo-terminal networks. As the name implies, two-terminal networks consist of a pair of terminals; the voltagepotential across the terminals and the current flow into the terminals characterizes the network. This approach isconsistent with our systems-level approach; we can characterize the behavior of what may be an extremelycomplex circuit by a relatively simple input-output relationship.

We already have some experience with two-terminal networks; when we determined equivalent resistances forseries and parallel resistor combinations, we treated the resistive network as a two-terminal network. For analysispurposes, the network was then replaced with a single equivalent resistance which was indistinguishable from theoriginal circuit by any external circuitry attached to the network terminals. In this chapter, we will formalize sometwo-terminal concepts and generalize our approach to include networks which contain both sources and resistors.

We will assume that the electrical circuit of interest can be subdivided into two sub-circuits, interconnected at twoterminals, as shown in Figure 4.5. Our goal is to replace circuit A in our overall system with a simpler circuitwhich is indistinguishable by circuit B from the original circuit. That is, if we disconnect circuit A from circuit Bat the terminals and replace circuit A with its equivalent circuit, the voltage v and the current i seen at theterminals of the circuits will be unchanged and circuit B’s operation will be unaffected. In order to make thissubstitution, we will need to use the principle of superposition in our analysis of circuit A – thus, circuit A must bea linear circuit. We are not changing circuit B in any way – circuit B can be either linear or nonlinear.

It should be emphasized that circuit A is not being physically changed. We are making the change conceptually inorder to simplify our analysis of the overall system. For example, the design of circuit B can now proceed with asimplified model of circuit A’s operation, perhaps before the detailed design of circuit A is even finalized. Whenthe designs of the two circuits are complete, they can be integrated and the overall system has a high probability offunctioning as expected.

CircuitA

(Linear)

CircuitBv

+

-

ia

b

Figure 4.5. Circuit composed of two, two-terminal sub-circuits.

In order to perform the above analysis, we will disconnect the two sub-circuits in Figure 4.5 at the terminals a – band determine the current-voltage relationship at the terminals of circuit A. We will generally refer to circuit A’scurrent-voltage relationship as its i-v characteristic. Our approach, therefore, is to look at circuit A alone, asshown in Figure 4.6, and determine the functional relationship between a voltage applied to the terminals and theresulting current. (Equivalently, we could consider that a current is applied at the terminals and look at the



resulting voltage.) Figure 4.6 is at first glance somewhat misleading – the terminals should not be considered to beopen-circuited, as a cursory look at the figure might indicate; we are determining the relationship between avoltage difference applied to the circuit and the resulting current flow. (Figure 4.6 indicates a current I flowinginto the circuit, which will, in general, not be zero.)

Systems-level interpretation:

When we determine the i-v characteristic for the circuit, we are determining the input-output relationship for asystem. Either the voltage or the current at the terminals can be viewed as the input to a system; the otherparameter is the output. The i-v characteristic then provides the output of the system as a function of the input.

Figure 4.6. Two-terminal representation of circuit.

Resistive Networks:

We have already (somewhat informally) treated purely resistive circuits as two-terminal networks when wedetermined equivalent resistances for series and parallel resistors. We will briefly review these concepts here in asystems context in terms of a simple example.



Example 4.6:

Determine the i-v characteristic for the circuit below.

Previously, we would use circuit reduction techniques to solve this problem. The equivalent resistance is

463)6()3(2eqR . Since iRv eq , the circuit’s i-v characteristic is iv 4 .

We would now, however, like to approach this problem in a slightly more general way and using a systems-levelview to the problem. Therefore, we will choose the terminal voltage, v, to be viewed as the input to the circuit. Bydefault, this means the current i will be our circuit’s output. (We could, just as easily define the current as theinput, in which case the voltage would become our output.) Thus, our circuit conceptually looks like a system asshown in the block diagram below.

Applying KCL to node c in the above circuit results in

36cc vv

i . Ohm’s law, applied to the 2 resistor,

results in ivv c 2 . Eliminating vc from the above two equations results in iv 4 , which is the same resultwe obtained using circuit reduction. The i-v characteristic for the above circuit is shown graphically below; theslope of the line is simply the equivalent resistance of the network.

In the above example, viewing the circuit as a general two-terminal network and using a more general systems-level approach to the problem results in additional work relative to using our previous circuit reduction approach.Viewing the circuit as a more general two-terminal network is, however, very profitable if circuit reduction



techniques are not applicable or if we allow the circuit to contain voltage or current sources. The latter topic isaddressed in the following subsection.

Two-terminal networks with sources:

When the network consists of resistive elements and independent sources, the circuit’s i-v characteristic can berepresented as a single equivalent resistance and a single source-like term. In general, however, we cannotdetermine this directly by using circuit reduction techniques. The overall approach and typical results areillustrated in the following examples.



Example 4.7:

Determine the i-v characteristic of the circuit below.

Although it is fairly apparent, by applying Ohm’s law across the series combination of resistors, that

SViRRv )( 21 , we will (for practice) use superposition to approach this problem. The voltage source Vs

will, of course, be one source in the circuit. We will use the voltage across the terminals a – b as a second source inthe circuit.

Killing the voltage source Vs results in the circuit to the left below; the resulting current is21

1 RRvi

. Killing

the “source” v results in the circuit to the right below; the resulting current is21

2 RRV

i S

. The total current

is, therefore,2121 RR

VRRvi S

or SViRRv )( 21 .

Plotting the above i-v characteristic results in the figure below.



Example 4.8:

Determine the i-v characteristic of the circuit below.

Although not the most efficient approach for this problem, we will again use superposition to approach theproblem. One source will, of course, be the current source IS. We will assume that our second source is thecurrent i at node a. Killing the current source IS results in the circuit to the left below; from this figure the voltagev1 can be seen to be )( 211 RRiv . Killing the current source i results in the figure to the right below; from thisfigure the voltage v2 is seen to be SIRv 12 (the dead current source results in an open circuit, so no current

flows through the resistor R2). The total voltage across the terminals is, therefore, SIRiRRv 121 )( .

The i-v characteristic for the circuit is, therefore, as shown in the figure below.



Notes on linear circuit i-v characteristics:

1. All two-terminal networks which contain only sources and resistors will have i-v relationships of the formshown in examples 1, 2, and 3. That is, they will be straight lines of the form bimv . The y-intercept term, b, is due to sources in the network; if there are no sources in the network, b = 0 and the i-vcharacteristic will pass through the origin.

2. Due to the form of the i-v characteristic provided in note 1 above, any two-terminal network can berepresented as a single source and a single resistor.

3. The form of the solution for examples 2 and 3 are the same. Thus, the circuit of example 2 is

indistinguishable from a similar circuit with a current source1RVS in parallel with the resistor R1.

Likewise, the circuit of example 3 is indistinguishable from a similar circuit with a voltage source 1RI S inseries with the resistor R1. The equivalent circuits are shown below.

1RVS



Section Summary:

Electrical circuits, sub-circuits, and components are often modeled by the relationship between voltage andcurrent at their terminals. For example, we are familiar with modeling resistors by Ohm’s law, which simplyrelates the voltage to the current at the resistor terminals. In Chapter 2, we used circuit reduction methods toextend this concept by replacing resistive networks with equivalent resistances which provided the samevoltage-current relations across their terminals. In this section, we continue to extend this concept to circuitswhich include sources.

For linear circuits, the voltage-current relationship across two terminals of the circuit can always berepresented as a straight line of the form bimv . If we plot this relationship with voltage on the verticalaxis and current on the horizontal axis, the slope of the line corresponds to an equivalent resistance seenacross the terminals, while the y-intercept of the line is the voltage across the terminals, if the terminals areopen-circuited. We will formalize this important result in section 4.5.

Exercises:

1. Determine the I-V characteristics of the circuit below, as seen at the terminals a-b.

Real Analog – Circuits 1Chapter 4.5: Thévenin's and Norton's Theorems


4.5: Thévenin’s and Norton’s TheoremsIn section 4.4, we saw that it is possible to characterize a circuit consisting of sources and resistors by the voltage-current (or i-v) characteristic seen at a pair of terminals of the circuit. When we do this, we have essentiallysimplified our description of the circuit from a detailed model of the internal circuit parameters to a simplermodel which describes the overall behavior of the circuit as seen at the terminals of the circuit. This simplermodel can then be used to simplify the analysis and/or design of the overall system.

In this section, we will formalize the above result as Thévenin’s and Norton’s theorems. Using these theorems, wewill be able to represent any linear circuit with an equivalent circuit consisting of a single resistor and a source.Thévenin’s theorem replaces the linear circuit with a voltage source in series with a resistor, while Norton’stheorem replaces the linear circuit with a current source in parallel with a resistor. In this section, we will applyThévenin’s and Norton’s theorems only to purely resistive networks. However, these theorems can be used torepresent any circuit made up of linear elements.

Consider the two interconnected circuits shown in Figure 4.7 below. The circuits are interconnected at the twoterminals a and b, as shown. Our goal is to replace circuit A in the system of Figure 4.7 with a simpler circuitwhich has the same current-voltage characteristic as circuit A. That is, if we replace circuit A with its simplerequivalent circuit, the operation of circuit B will be unaffected. We will make the following assumptions about theoverall system:

Circuit A is linear Circuit A has no dependent sources which are controlled by parameters within circuit B Circuit B has no dependent sources which are controlled by parameters within circuit A

Figure 4.7. Interconnected two-terminal circuits.

In section 4.3, we determined i-v characteristics for several example two-terminal circuits, using the superpositionprinciple. We will follow the same basic approach here, except for a general linear two-terminal circuit, in orderto develop Thévenin’s and Norton’s theorems.

Thévenin’s Theorem:

First, we will kill all sources in circuit A and determine the voltage resulting from an applied current, as shown inFigure 4.8 below. With the sources killed, circuit A will look strictly like an equivalent resistance to any external



circuitry. This equivalent resistance is designated as RTH in Figure 4.8. The voltage resulting from an appliedcurrent, with circuit A dead is:

iRv TH 1 (4.3)

Figure 4.8. Circuit schematic with dead circuit.

Now we will determine the voltage resulting from re-activating circuit A’s sources and open-circuiting terminals aand b. We open-circuit the terminals a-b here since we presented equation (4.3) as resulting from a currentsource, rather than a voltage source. The circuit being examined is as shown in Figure 4.9. The voltage vOC is the“open-circuit” voltage.

Figure 4.9. Open-circuit response.

Superimposing the two voltages above results in:

OCvvv 1 (4.4)

or

OCTH viRv (4.5)

Equation (4.5) is Thévenin’s theorem. It indicates that the voltage-current characteristic of any linear circuit (withthe exception noted below) can be duplicated by an independent voltage source in series with a resistance RTH,known as the Thévenin resistance. The voltage source has the magnitude vOC and the resistance is RTH, where vOC

is the voltage seen across the circuit’s terminals if the terminals are open-circuited and RTH is the equivalentresistance of the circuit seen from the two terminals, with all independent sources in the circuit killed. Theequivalent Thévenin circuit is shown in Figure 4.10



Procedure for determining Thévenin equivalent circuit:

1. Identify the circuit and terminals for which the Thévenin equivalent circuit is desired.2. Kill the independent sources (do nothing to any dependent sources) in circuit and determine the

equivalent resistance RTH of the circuit. If there are no dependent sources, RTH is simply the equivalentresistance of the resulting resistive network. Otherwise, one can apply an independent current source atthe terminals and determine the resulting voltage across the terminals; the voltage-to-current ratio is RTH.

3. Re-activate the sources and determine the open-circuit voltage VOC across the circuit terminals. Use anyanalysis approach you choose to determine the open-circuit voltage.



Example 4.9:

Determine the Thévenin equivalent of the circuit below, as seen by the load, RL.

We want to create a Thévenin equivalent circuit of the circuit to the left of the terminals a-b. The load resistor, RL,takes the place of “circuit B” in Figure 1.

The circuit has no dependent sources, so we kill the independent sources and determine the equivalent resistanceseen by the load. The resulting circuit is shown below.

From the above figure, it can be seen that the Thévenin resistance RTH is a parallel combination of a 3 resistor

and a 6 resistor, in series with a 2 resistor. Thus,

4236)3)(6(

THR .

The open-circuit voltage vOC is determined from the circuit below. We (arbitrarily) choose nodal analysis todetermine the open-circuit voltage. There is one independent voltage in the circuit; it is labeled as v0 in the circuitbelow. Since there is no current through the 2 resistor, vOC = v0.

Applying KCL at v0, we obtain: VvvvVv

A OC 6036

62 0

00

. Thus, the Thévenin equivalent

circuit is on the left below. Re-introducing the load resistance, as shown on the right below, allows us to easilyanalyze the overall circuit.

+-6V

+-6V RL



Norton’s Theorem:

The approach toward generating Norton’s theorem is almost identical to the development of Thévenin’s theorem,except that we apply superposition slightly differently. In Thévenin’s theorem, we looked at the voltage responseto an input current; to develop Norton’s theorem, we look at the current response to an applied voltage. Theprocedure is provided below.

Once again, we kill all sources in circuit A, as shown in Figure 4.8 above but this time we determine the currentresulting from an applied voltage. With the sources killed, circuit A still looks like an equivalent resistance to anyexternal circuitry. This equivalent resistance is designated as RTH in Figure 4.8. The current resulting from anapplied voltage, with circuit A dead is:

THRvi 1 (4.6)

Notice that equation (4.6) can be obtained by rearranging equation (4.3)

Now we will determine the current resulting from re-activating circuit A’s sources and short-circuiting terminals aand b. We short-circuit the terminals a-b here since we presented equation (4.4) as resulting from a voltagesource. The circuit being examined is as shown in Figure 4.11. The current iSC is the “short-circuit” current. It istypical to assume that under short-circuit conditions the short-circuit current enters the node at a; this isconsistent with an assumption that circuit A is generating power under short-circuit conditions.

Figure 4.11. Short-circuit response.

Employing superposition, the current into the circuit is (notice the negative sign on the short-circuit current,resulting from the definition of the direction of the short-circuit current opposite to the direction of the current i)

SCiii 1 (4.7)

so

SCTH

iRvi (4.8)

Equation (4.8) is Norton’s theorem. It indicates that the voltage-current characteristic of any linear circuit (withthe exception noted below) can be duplicated by an independent current source in parallel with a resistance. Thecurrent source has the magnitude iSC and the resistance is RTH, where iSC is the current seen at the circuit’sterminals if the terminals are short-circuited and RTH is the equivalent resistance of the circuit seen from the two



terminals, with all independent sources in the circuit killed. The equivalent Norton circuit is shown in Figure4.12.

Figure 4.12. Norton equivalent circuit.

Procedure for determining Norton equivalent circuit:

1. Identify the circuit and terminals for which the Norton equivalent circuit is desired.2. Determine the equivalent resistance RTH of the circuit. The approach for determining RTH is the same for

Norton circuits as Thévenin circuits.3. Re-activate the sources and determine the short-circuit current iSC across the circuit terminals. Use any

analysis approach you choose to determine the short-circuit current.



Example 4.10:

Determine the Norton equivalent of the circuit seen by the load, RL, in the circuit below.

This is the same circuit as our previous example. The Thévenin resistance, RTH, is thus the same as calculatedpreviously: RTH = 4. Removing the load resistance and placing a short-circuit between the nodes a and b, asshown below, allows us to calculate the short-circuit current, iSC.

Performing KCL at the node v0, results in:

AvVvv

236

62

000

so

Vv 30

Ohms’ law can then be used to determine iSC:

AViSC 5.123

and the Norton equivalent circuit is shown on the left below. Replacing the load resistance results in theequivalent overall circuit shown to the right below.



Exceptions:

Not all circuits have Thévenin and Norton equivalent circuits. Exceptions are:

1. An ideal current source does not have a Thévenin equivalent circuit. (It cannot be represented as avoltage source in series with a resistance.) It is, however, its own Norton equivalent circuit.

2. An ideal voltage source does not have a Norton equivalent circuit. (It cannot be represented as a currentsource in parallel with a resistance.) It is, however, its own Thévenin equivalent circuit.

Source Transformations:

Circuit analysis can sometimes be simplified by the use of source transformations. Source transformations areperformed by noting that Thévenin’s and Norton’s theorems provide two different circuits which provideessentially the same terminal characteristics. Thus, we can write a voltage source which is in series with aresistance as a current source in parallel with the same resistance, and vice-versa. This is done as follows.

Equations (4.5) and (4.8) are both representations of the i-v characteristic of the same circuit. Rearrangingequation (4.5) to solve for the current i results in:

TH

OC

TH Rv

Rvi (4.9)

Equating equations (4.8) and (4.9) leads to the conclusion that

TH

OCSC R

vi (4.10)

Likewise, rearranging equation (4.8) to obtain an expression for v gives:

THSCTH RiRiv (4.11)

Equating equations (4.11) and (4.5) results in:

THSCOC Riv (4.12)

which is the same result as equation (4.10).

Equations (4.10) and (4.12) lead us to the conclusion that any circuit consisting of a voltage source in series with aresistor can be transformed into a current source in parallel with the same resistance. Likewise, a current sourcein parallel with a resistance can be transformed into a voltage source in series with the same resistance. The valuesof the transformed sources must be scaled by the resistance value according to equations (4.10) and (4.12). Thetransformations are depicted in Figure 4.13.



RVS

RI S

Figure 4.13. Source transformations.

Source transformations can simplify the analysis of some circuits significantly, especially circuits which consist ofseries and parallel combinations of resistors and independent sources. An example is provided below.



Example 4.11:

Determine the current i in the circuit shown below.

We can use a source transformation to replace the 9V source and 3 resistor series combination with a 3A sourcein parallel with a 3 resistor. Likewise, the 2A source and 2 resistor parallel combination can be replaced with a4V source in series with a 2 resistor. After these transformations have been made, the parallel resistors can becombined as shown in the figure below.

The 3A source and 2 resistor parallel combination can be combined to a 6V source in series with a 2 resistor,as shown below.

The current i can now be determined by direct application of Ohm’s law to the three series resistors, so that

AVVi 25.0242

46

.

Voltage – Current Characteristics of Thévenin and Norton Circuits:

Previously, in section 4.4, we noted that the i-v characteristics of linear two-terminal networks containing onlysources and resistors are straight lines. We now look at the voltage-current characteristics in terms of Théveninand Norton equivalent circuits.



Equations (4.5) and (4.8) both provide a linear voltage-current characteristic as shown in Figure 4.14. When thecurrent into the circuit is zero (open-circuited conditions), the voltage across the terminals is the open-circuitvoltage, vOC. This is consistent with equation (4.5), evaluated at i = 0:

OCOCTHOCOCTH vvRviRv 0 .

Likewise, under short-circuited conditions, the voltage differential across the terminals is zero and equation (4.8)readily provides:

scscTH

scTH

SC iiR

iRv

i 0

which is consistent with Figure 4.14.

Figure 4.14. Voltage-current characteristic for Thévenin and Norton equivalent circuits.

Figure 4.14 is also consistent with equations (4.10) and (4.12) above, since graphically the slope of the line is

obviouslySC

OCTH i

vR .

Figure 4.14 also indicates that there are three simple ways to create Thévenin and Norton equivalent circuits:

1. Determine RTH and vOC. This provides the slope and y-intercept of the i-v characteristic. This approach isoutlined above as the method for creating a Thévenin equivalent circuit.

2. Determine RTH and iSC. This provides the slope and x-intercept of the i-v characteristic. This approach isoutlined above as the method for creating a Norton equivalent circuit.

3. Determine vOC and iSC. The equivalent resistance RTH can then be calculated fromSC

OCTH i

vR to

determine the slope of the i-v characteristic. Either a Thévenin or Norton equivalent circuit can then becreated. This approach is not commonly used, since determining RTH – the equivalent resistance of thecircuit – is usually easier than determining either vOC or iSC.



Note:

It should be emphasized that the Thévenin and Norton circuits are not independent entities. One can always bedetermined from the other via a source transformation. Thévenin and Norton circuits are simply two differentways of expressing the same voltage-current characteristic.

Section Summary:

Thévenin’s theorem allows us to replace any linear portion of a circuit with equivalent circuit consisting of avoltage source in series with a resistance. This circuit is called the Thévenin equivalent, and provides the samevoltage-current relationship at the terminals as the original circuit. The voltage source in the equivalentcircuit is the same as the voltage which would be measured across the terminals of the original circuit, if thoseterminals were open-circuited. The resistance in the equivalent circuit is called the Thévenin resistance, it isthe resistance that would be seen across the terminals of the original circuit, if all sources in the circuit werekilled.

Norton’s theorem allows us to replace any linear portion of a circuit with equivalent circuit consisting of acurrent source in parallel with a resistance. This circuit is called the Norton equivalent, and provides the samevoltage-current relationship at the terminals as the original circuit. The current source in the equivalentcircuit is the same as the current which would be measured across the terminals of the original circuit, if thoseterminals were short-circuited. The resistance in the equivalent circuit is the resistance that would be seenacross the terminals of the original circuit, if all sources in the circuit were killed; it is the same as theThévenin resistance.

Thevenin and Norton’s theorems allow us to perform source transformations when analyzing circuits. Thisapproach simply allows us to replace any voltage source which is in series with a resistance with a currentsource in parallel with the same resistance, and vice-versa. The relationship between the voltage and currentsources used in these transformations are provided in equations (4.10) and (4.12).



Exercises:

1. Replace everything except the 1A current source with its Thevenin equivalent circuit and use the result to findV1.

41A

+

-8V

4 +

-

V1

2. Replace everything except the 1A current source with its Norton equivalent circuit and use the result to findV1.

3. Determine a Norton equivalent circuit for the circuit below.

Real Analog – Circuits 1Chapter 4.6: Maximum Power Transfer


4.6: Maximum Power TransferIt is often important for our electrical system to transfer as much power as possible to some related system. Forexample, in an audio system it is important that the amplifier transfer as much power as possible to theloudspeakers. Otherwise, the amplifier generates power which is not used for any productive purpose1 and theefficiency of the overall system suffers.

In this section, we will develop design guidelines which will ensure that the maximum possible amount of power istransferred from our electrical circuit to the load that the circuit is driving. These guidelines will be based onThevenin’s theorem.

Consider the system shown in Figure 4.15. The overall system consists of an electrical circuit which is being usedto drive a load. Physically, the load can be either another electrical system or some electromechanical system suchas an electric motor or a loudspeaker. We will model the load as an electrical resistance, RL, though the principlespresented here are applicable to more general loading conditions.

Figure 4.15. General electrical network – load combination.

We will replace our electrical system with its Thevenin equivalent in order to analyze the power delivered by thecircuit to the load. The overall circuit that we are analyzing is now modeled as shown in Figure 4.16.

Figure 4.16. Electric circuit – load combination. Electric circuit modeled by its Thevenin equivalent.

From Figure 4.16, the voltage delivered to the load can be readily determined from a voltage divider relation:

THL

LOCL RR

RVV

(4.13)

1 Other than, perhaps, heating the room it is in.



Thus, the power delivered to the load is

222

THL

L

L

OC

L

LL RR

RRV

RV

P (4.14)

Figure 4.17 shows a plot of the power delivered to the load, as a function of the load resistance. The powerdelivered to the load is zero when the load resistance is zero (since there is no voltage drop across the load underthis condition) and goes to zero as the load resistance approaches infinity (since there is no current provided tothe load under this condition). At some value of RL the power transfer will be maximized -- our goal will be todetermine the value for RL which maximizes the power delivered to the load.

Figure 4.17. Delivered power vs. load resistance.

The maximum value of power on the curve shown in Figure 4.17 can be determined by differentiating equation(4.14) with respect to the load resistance RL and setting the result to zero. This leads to:

0)(

)(2)(4

22

THL

LTHLTHLOC

L

L

RRRRRRRV

RP

(4.15)

The above condition is satisfied if the numerator of equation (4.15) is zero, so our condition becomes

)(2)( 2LThLTHL RRRRR

Dividing both sides by LTh RR results in

LTHL RRR 2

or

THL RR (4.16)



Thus, maximum power transfer takes place when the load resistance and the Thevenin resistance of the circuitsupplying the power are equal. The above result is sometimes called the maximum power transfer theorem. Whenthe conditions of the maximum power transfer theorem are met, the total power delivered to the load is:

TH

OC

TH

OC

L RV

R

V

P4

2 2

2

(4.17)

This is one half of the total power generated by the circuit, half the power is absorbed in the resistance RTH.

Conclusion:

The power delivered to the load is maximized if the load resistance is equal to the Thevenin resistance of thecircuit supplying the power. When this condition is met, the circuit and the load are said to be matched. Whenthe load and the circuit are matched, 50% of the power generated in the circuit can be delivered to the load –under any other circumstances, a smaller percentage of the generated power will be provided to the load.

Practical Power Supplies:

Practical power supplies were discussed in section 2.4. It was seen that the presence of an internal resistance in avoltage or current source limited the power that could be delivered to a circuit connected to the source. Practicalpower supplies are a special case of the results presented above; we use them below as examples of the applicationof the above principles.

In section 2.4, practical voltage sources were modeled as an ideal voltage source VS in series with some internalresistance RS, as shown in Figure 4.18. This corresponds exactly to a Thevenin circuit with VOC = VS and RTH = RS.The practical voltage source provides maximum power to a circuit connected to it when the input resistance of thecircuit (the equivalent resistance of the circuit, seen at the terminals to which the power source is connected) isequal to the internal resistance of the voltage source. Under these circumstances, the power delivered to thecircuit is:

S

S

RV

P4

2

The same amount of power is converted to heat within the power supply; this is the reason many power suppliescontain a fan to actively disperse this heat to the atmosphere. If the circuit’s input resistance is not equal to thesource resistance, less power is transmitted to the circuit and a correspondingly greater amount is dissipatedwithin the power supply.



Figure 4.18. Practical voltage source model.

Practical current sources were modeled in section 2.4 as an ideal current source IS in parallel with some internalresistance RS, as shown in Figure 4.19. This corresponds directly to a Norton equivalent circuit with ISC = IS andRTH = RS. The current source provides maximum power to a circuit connected to it when the input resistance ofthe circuit is equal to the internal resistance of the source. A source transformation in conjunction with equation(5) indicates that the power delivered to the circuit by the current source is:

4

2SS IRP

Again, a reduced percentage of the power generated by the source will be delivered to the circuit when the circuitand source are not well matched.

Figure 4.19. Non-ideal current source model.

Often, it may not be feasible to match the load with the power supply. For example, when we are testing circuitsin our lab assignments we do not generally attempt to maximize the power delivered to the circuit – this is typicalwhen prototype circuits are being tested. One simply recognizes that excessive power is being dissipated withinthe power supply and that the overall system is not functioning efficiently. If, however, the power supply andassociated circuit are being designed as part of an integrated overall system one will generally attempt to matchthe power supply to the rest of the system.



One problem which can occur during circuit testing is that extremely poorly matched power supply-loadcombinations may result in so much power being dissipated within the power supply that insufficient power isavailable to drive the load. This can result in the load apparently behaving abnormally, unless power deliveryeffects are considered.

Section Summary:

The maximum power that a circuit can deliver to a load resistor occurs when the load resistance is equal to theThévenin equivalent resistance of the circuit.

Exercises:

1. Determine the resistance R which will absorb the maximum power from the 7V source.

4. introduction and chapter objectives - digilentincreal analog - circuits 1 chapter 4: systems and...

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